The effective Borel hierarchy

• Autorzy: M. Vanden Boom
• Języki publikacji: EN

Abstrakty

EN

Let K be a subclass of Mod(𝓛) which is closed under isomorphism. Vaught showed that K is $Σ_α$ (respectively, $Π_α$) in the Borel hierarchy iff K is axiomatized by an infinitary $Σ_α$ (respectively, $Π_α$) sentence. We prove a generalization of Vaught's theorem for the effective Borel hierarchy, i.e. the Borel sets formed by union and complementation over c.e. sets. This result says that we can axiomatize an effective $Σ_α$ or effective $Π_α$ Borel set with a computable infinitary sentence of the same complexity. This result yields an alternative proof of Vaught's theorem via forcing. We also get a version of the pull-back theorem from Knight et al. which says if Φ is a Turing computable embedding of K ⊆ Mod(𝓛) into K' ⊆ Mod(ℒ'), then for any computable infinitary sentence φ in the language 𝓛, we can find a computable infinitary sentence φ* in 𝓛' such that for all 𝓐 ∈ K, 𝓐 ⊨ φ* iff Φ(𝓐 ) ⊨ φ, where φ* has the same complexity as φ.

Kategorie tematyczne

• 03D55: Hierarchies
• 03C75: Other infinitary logic
• 03E15: Descriptive set theory
• 03C57: Effective and recursion-theoretic model theory

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2007
• Tom: 195
• Numer: 3
• Strony: 269-289

Daty

wydano

2007

Twórcy

autor

M. Vanden Boom

• Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, U.S.A.

Identyfikatory

DOI

10.4064/fm195-3-4